influences of t-stress on constraint effect in mismatched ... · effect in mismatched. modified ....
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INFLUENCES OF T-STRESS ON CONSTRAINT EFFECT IN MISMATCHED MODIFIED BOUNDARY LAYER MODEL FOR CREEP CRACK
Yanwei Dai Department of Engineering Mechanics, AML,
Tsinghua University, Beijing, China
e-mail:[email protected]
Yinghua Liu* Department of Engineering Mechanics, AML,
Tsinghua University, Beijing, China
e-mail: [email protected] Haofeng Chen
Department of Mechanical & Aerospace Engineering, University of Strathclyde,
Glasgow, G1 1XJ, UK e-mail: [email protected]
ABSTRACT
Constraint effect plays an important role in assessing the
stress field and the growth rate of creep crack in components
under high temperature. The mismatched modified boundary
layer (MMBL) model is extended to creep crack in this paper.
For the MMBL model, the Q-parameters for different mismatch
factors are studied under different T-stresses. The variation of the
dimensionless T-stress in creep zone is given. The variations of
open stresses with creep time for different mismatch factors are
presented under different T-stresses. The comparisons of Q-
parameter between homogeneous material and mismatched
materials are made. The influences of mismatch factor on the
constraint parameter are discussed. The influence of creep
exponent on the open stress is also discussed.
NOMENCLATURE
1,2,3ia i Coefficients of Q-T relationship
AB Creep constant of power law creep for base
metal
AC Unified constraint parameter given by Ma et
al.
AW Creep constant of power law creep for base
metal
Constraint parameter given by Chao et al.
C(t) C-integral for transient creep
C* C*-integral for extensive creep
E Young’s modulus
J J-integral
KI Stress intensity factor of mode I
In Integral constant for HRR field
MP Strength mismatch factor for plastic material
MC Creep mismatch factor
n Creep exponent
Q Constraint parameter
r Distance from crack tip
T T-stress
ux Displacement in x-direction
uy Displacement in y-direction
v Poisson’s ratio
0 Reference creep strain rate
Yielding stress
ij
Stress component
ij Angular distribution function
22 Open stress
HRR
22 Open stress of HRR field
SSY, 0
22
T Open stress of small scale yielding
SSC, 0
22
T Open stress of small scale creep
Polar angle
*
2A
0
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1 INTRODUCTION
The mismatch effect of creep crack can affect the crack tip
stress field greatly. The accurate prediction and better
understanding of creep crack tip stress field is a basis to evaluate a mismatched creep crack. The so-called mismatch effect is
mainly caused by the difference of materials. For an elastoplastic
weldment, the mismatch effect is always characterized by a
mismatch factor pM , which is defined by the ratio of yielding
stress between weld metal and base metal, shown as below
yW
p
yB
M
(1)
where yW is the yielding stress of weld part and yB is the
yielding stress of base part. A lot of studies have been focused
on the mismatch effect of elastoplastic material. The estimation
of J-integral for a typical dissimilar material was presented by
Haddi and Weichert [1]. The mismatch effect on the perfectly
elastoplastic welding specimens was investigated by Kim and
Schwalbe [2, 3]. Song et al. [4] presented the so-called
mismatched limit load and J-integral approximation of surface
flaw in a tensional plate. Except for the discussions of stress and
fracture parameter estimation of crack tip for an elastoplastic
material, there were also some researches on the constraint effect
of mismatched weldments, e.g. Zhang et al. [5] gave a two
parameter J-M method to characterize the constraint effect for an
interfacial crack. Similar investigations can be also seen in Refs.
[6, 7].
The mismatch effect also exists in creeping weldments.
However, the definition of mismatch factor for a creep welding
component is rather different from an elastoplastic weldment. As
usual, the definition of mismatch factor for creeping solids with
an elastic power-law constitutive equation can be presented as
1/
BC
W
n
AM
A
(2)
where BA , WA and n are respectively the creep coefficients for
base metal, weld metal and creep exponent for power-law creep
equation given as below
B B
nA (3)
W W
nA (4)
in which B and W are the creep strain rates of base metal
and weld metal, respectively.
For creeping weldments, a lot of discussions were focused
on the creep stress distribution, e.g. Lee et al. [8] gave a study on
the quantification of creeping stress of a welded branched pipe.
Han et al. [9] also presented the creeping stress distribution of a
welded branched pipe junction with a heat affected zone (HAZ).
There were also some other related works on creeping
weldments [10-12]. Besides the stress analysis of creeping
components, the constraint effect for creep crack was also found
to have a great influence on the evaluation of stress field [13].
Recently, some constraint parameters were proposed to
characterize the constraint effect for creep crack, e.g. Q-
parameter [13-15], *
2A -parameter [16], R-parameter [17], R*-
parameter [18] and Ac-parameter [19]. However, there are so far
few discussions on quantifying the constraint effect of mismatch
creep crack. Dai et al. [20] presented the M*-parameter as a
constraint parameter for characterization of material constraint
effect of creep crack, where a mismatched modified boundary
layer (MMBL) model was used.
In fact, the boundary layer model was used widely in
elasticity or plasticity materials [21, 22]. For creeping materials,
Matvienko et al. [23] used the modified boundary layer (MBL)
model to investigate the in-plane constraint parameter *
2A and
out-of-plane constraint parameter zT under different T-stresses.
Though some works like Refs. [20, 23] were presented, the
influence of T-stress on the mismatched creep crack has not been
investigated thoroughly. Especially, the influence of T-stress on
the Q-parameter under creeping regime is still unknown yet. In
this paper, the influence of T-stress on constraint effect of
mismatch creep crack in MMBL model is studied.
2 MISMATCHED MODIFIED BOUNDARY LAYER MODEL AND NUMERICAL PROCEDURE
Fig. 1 FE grid of MMBL model with creep crack
The MMBL model is carried out with a circular disc shown
in Fig. 1, where the x and y-axis coordinates are also presented.
A half model is used here because of the symmetry of the MMBL
model. The boundary conditions can be referred to Ref. [20],
which are governed by analytical solutions for displacements of
mode I crack with elastic field at the outer boundary. The
displacements on the outer boundary of the MMBL model can
be written as
I
2
1 3 4 cos cos2 2
1 cos
x
K ru v v
E
Tv r
E
(5)
I 1 3 4 cos sin2 2
1 sin
y
K ru v v
E
Tv v r
E
(6)
x
y
r
Symmetric boundary
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where IK is the stress intensity factor (SIF) of linear elasticity
for mode I crack, v is the Poisson’s ratio and r and are
the polar coordinates also shown in Fig. 1. T is the applied T-
stress on the outer boundary of the MBL model. According to the
MBL model, the small scale yielding can be obtained if the
circular radius is large enough. Considering the similarity of
HRR singularity [24] between power law elastoplastic field and
power law elastic-creep field, the small scale creep can be
obtained if the circular radius is large enough and the T-stress is
not high, which is demonstrated by Dai et al. [25].
Fig. 2 FE mesh of crack tip for MMBL model
The finite element (FE) code ABAQUS is adopted here to
perform the numerical analyses. The CPE8R is selected as the
element type to simulate the crack tip. The detailed crack tip
mesh can be seen in Fig. 2. The height of the weld metal is taken
as 1.0 mm, and a half-length of 0.5 mm is shown in Fig. 2. The
total element number of the MMBL in this paper is 1844. The
material constants of P92 steel [26] for elastic power-law creep
are adopted here, which can be seen in Table 1. For the analyzed
cases, the creep exponents are kept as 5.23 and 7, respectively.
The elastic modulus E, yielding stress 0 and Poisson’s ratio
are taken as 125000 MPa, 180 MPa and 0.3, respectively.
Table 1 Creep material constants used in the calculation
AB(MPa-nh-1) AW(MPa-n
h-1) MC n
2.64E-16 2.64E-14 0.41 5.23
2.64E-16 2.64E-16 1.00 5.23
2.64E-16 2.64E-18 2.41 5.23
3.20E-19 1.60E-18 0.79 7
3.20E-19 3.20E-19 1.00 7
3.20E-19 6.40E-20 1.26 7
The applied far-field boundary conditions with mode I
crack are used to predict the near crack tip stress field of MMBL
model. To verify the accuracy of the applied boundary
conditions, a comparison is made with the predicted elastic crack
tip field in Table 2. It can be seen that the relative errors between
the applied SIF and the predicted SIF are much lower than 10%,
which is always chosen as the upper error bound for the MBL
model estimation. During the whole analysis, the applied SIF is
kept as 100 MPa mm1/2.
Table 2 A comparison between the applied and predicted SIF
Applied SIF
(MPa mm1/2)
Predicted SIF
(MPa mm1/2)
Relative
Error
100 101.9 1.90%
150 152.8 1.86%
200 203.7 1.85%
3 VARIATIONS OF T-STRESSES
In general case, the T-stress of near field for creep crack can
be presented as
o for =0xx yyT (7)
where xx and yy are the stress of creep crack front in x-
direction and y-direction, respectively. The variations of
dimensionless T-stress with r under the mismatch conditions are
presented in Fig. 3. The T-stress in the legend represents the
applied T-stress on the outer boundary. It can be found that the
variations of T-stress for near field is dependent on mismatch
factors, and dimensionless T-stress under the same mismatch
factor presents the same variation tendency. Under the lower
match condition, the dimensionless T-stress coincides well for
different T-stresses of near crack tip where r<1 mm. If the r
exceeds the creep zone, the dimensionless T-stress of near crack
tip here agrees well with the applied T-stress on the outer
boundary. It implies that the stress field of MMBL on the
mismatch condition has the same nature as the homogeneous
condition, and it also can support that the MMBL in creep range
is still valid here.
Fig. 3 Variations of dimensionless T-stresses under the under-
mismatch condition for different T-stresses
Weld metal
Base metal
Crack tip
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4 SELF-SIMILAR VERIFICATION OF MISMATCH CREEP CRACK STRESS FIELD
The self-similar property for the mismatched creep crack tip
has been verified by Dai et al. [20] with a normalized stress field.
In this paper, we present the creep strain distributions of crack
for different mismatch factors under T=0. Herein, the creep zone
size is shown in Fig. 4 at creep time of 100000 hours with
equivalent creep strain (CEEQ) of 1E-4. It can be seen that the
creep zone size under the over-match condition with mismatch
factor of 2.41 is much smaller compared with that under the
even-match and the under-match conditions. In weld metal, the
creep zone size under the under-match condition is much higher
than that under the over-match condition. The above reason is
that the material constraint under the over-match condition has
the higher constraint value than that under the under-match and
even-match conditions, and the higher constraint level restricts
the creep zone in a small region. By the comparison of creep
zone at different creep time, the distribution of CEEQ can
explain clearly the self-similar property of the MMBL model.
Fig. 4 CEEQ zone under different mismatch factors
5 T-STRESS EFFECT ON CONSTRAINT OF MISMATCHED CREEP CRACK
As a matter of fact, the T-stress term is obtained from the
Williams’ expansions [27]. For the MMBL model, the region
near the crack tip is dominated by the creep zone compared with
that of far field controlled by the elastic zone, which was
demonstrated by Riedel and Rice [28]. Because of the analogy
between the HRR field [29, 30] and the elastic power-law field,
the stress field of creep zone for MBL model obeys the HRR
singularity. According to the HRR field, the stress field of creep
crack under the dominance of C(t) in transient creep, or C*-
integral in extensive creep, can be written as
1/ 1
0
0 0
( ), ,
n
ij ij
n
C tr n
I r
(8)
1/ 1
*
0
0 0
,
n
ij ij
n
Cr
I r
(9)
where 0 , 0 , nI , r and ij are yielding stress,
reference creep strain rate, integral constant, distance from crack
tip and angular function, respectively.
If the constraint effect is taken into consideration, the stress
field for elastic power-law creep, can be written as [13]
1/ 1
*
0 0
0 0
, ,
n
ij ij ij
n
Cr n Q
I r
(10)
where Q is the creeping constraint parameter similar to
elastoplastic constraint parameter given by Shih et al. [31], and
ij is the Kronecker’s delta. Generally, the constraint parameter
Q can be calculated as follows [31]:
HRR
22 22
0
Q
(11)
SSY, =0
22 22
0
T
Q
(12)
where HRR
22 and SSY, =0
22
T are the open stress (or tangential
stress) of HRR field under small scale yielding with T=0. As for
the elastoplastic material, the relationship between the Q-
parameter and T-stress can be described by [31]
1 2 2
1 2 3
0 0 0
T T TQ a a a
(13)
where 1a , 2a and 3a are coefficients depending on loading
level and material properties.
As the stress field of creep crack can be affected by the C(t)-
integral, the C(t)-integrals under different mismatch factors are
firstly discussed in Fig. 5. It implies that the C(t)-integral on the
under-match condition is higher than the C(t)-integral on the
even-match condition for short creep time. If the creep time is
long enough, the C(t)-integral approaches to be nearly the same.
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Fig. 5 Variations of C*-integrals with creep time under different
mismatch factors
With Eqs. (7)-(8), the open stresses of analytical HRR field
at 100000 hours are shown in Fig. 5. It can be seen that the open
stress 22 under the even-match condition with T=0 coincides
with analytical HRR field very well. However, the open stress on
the under-match condition with mismatch factor of 0.41 is much
lower than that under the even-match condition. The open stress
on the over-match condition is larger than that under the even-
match and under-match conditions. As the high constraint level
on the over-match condition has a much smaller creep zone, the
open stress near the crack tip on the over-match condition
deviates from the HRR field greatly.
Fig. 6 Comparison of open stresses at long creep time for
different mismatch factors with T=0
As the C(t)-integral under different mismatch factors at long
creep time does not differ much from each other, the open
stresses of analytical field presented in Fig. 6 agree well with
each other. It should be pointed out that the differences between
the open stresses of the analytical HRR field are significant if the
creep time is very short.
To have a better understanding of the T-stress in the stress
field of creeping crack tip, the open stresses of MMBL model
under different mismatch factors with various T-stresses are
presented in Figs. 7-9. It can be seen that the open stress of creep
crack tip coincides with the HRR field perfectly only when T=0
under even-match condition. It implies that the stress field with
T=0 can be used as the reference stress field to characterize the
constraint effect of creep crack as Eq. (12). Under the even-
match condition, the open stress of creep crack tip for negative
T-stress is lower than that of the HRR field, and also lower than
that of creep crack tip for positive T-stress, as shown in Fig. 7.
Under the under-match condition presented in Fig. 8, the
open stresses of creep crack tip are lower than those of the HRR
field for both negative T-stress and positive T-stress. The
difference of open stresses between positive T-stress and
negative T-stress is slight. Under the over-match condition
shown in Fig. 9, the conclusion is quite similar to that under the
under-match condition, however, the difference is that the open
stress in creep zone on the over-match condition is much higher
than that of the HRR field.
Fig. 7 Comparison of open stresses at long creep time on the
even-match condition for different T-stresses
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Fig. 8 Comparison of open stresses at long creep time on the
under-match condition for different T-stresses
Fig. 9 Comparison of open stresses at long creep time on the
over-match condition for different T-stresses
Based on the analysis of open stress, Eq. (11) is first used
to characterize the constraint effect of mismatched creep crack
in the MMBL model (see Fig. 10). It can be seen that positive
values of constraint parameter Q of creep crack are obtained on
the over-match condition, while negative values are gotten on the
under-match condition.
Fig. 10 Variations of Q for different mismatch factors and T-
stresses
Fig. 11 Variations of Q for different mismatch factors and T-
stresses
Instead of Eqs. (11)-(12), a formula to characterize the
constraint effect of mismatched creep crack is proposed as
SSC, =0
22 22
0
TM MQ
(14)
where 22 M means the open stress of mismatched creep
crack with T≠0 and SSC, =0
22
T M represents the open stress of
mismatched creep crack with T=0 under small scale creep. With
the definition of Eq. (14), the constraint effect caused by loading
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condition can be obtained (see Fig 11). According to Dai et al.
[20], Eq. (14) is the constraint effect caused by generalized
geometry (loading mode and geometric size), and the constraint
effect of material is characterized by the material constraint
parameter M*. It can be seen that for a mismatched creep crack
in the MMBL model, the material mismatch constraint effect
plays a more significant role than the geometric constraint effect.
The role of T-stress can affect the geometric constraint only.
Creep is time-dependent, and the behavior of creep
relaxation is the most important difference between creep and
plasticity. Figs. 12-13 present the variations of open stress with
time on the over-match condition and the under-match condition.
The open stresses given here are obtained at the same distance
from crack tip with r=0.1 mm in the creep zone. It can be seen
that the variations of open stress on the over-match and the
under-match conditions are rather different. Under the over-
match condition, the influence of T-stress on open stress of creep
crack tip is not significant, however, there are blunting near the
crack tip. Under the under-match condition, the influence of T-
stress on open stress is slight. The above reason is that the creep
relaxation under the over-match condition is significant as the
creep crack tip has a higher stress level. Under the under-match
condition, the influence of T-stress on the stress level of creep
crack tip is not significant like that under the over-match
condition.
Fig. 12 Variations of dimensionless open stress under the over-
mismatch condition for different T-stresses
Fig. 13 Variations of dimensionless open stress under the under-
mismatch condition for different T-stresses
6 INFLUENCE OF CREEP EXPONENT
As the creep exponent is also significant on the influence of
the creep behavior, the investigation on the influence of creep
exponent. The creep constant for n=7 is used here where the
specific constants can be seen from Table 1. Herein, three
different mismatch factors, i.e. m=0.83, 1.00 and 1.39, have the
different creep coefficients compared with n=5.23. The
variations of dimensionless open stress under different mismatch
factors for different creep exponent are presented in Fig.14. The
interesting thing is that the open stresses here have the same
tendencies for m=1.0 though the creep coefficients and creep
exponents differ much. The open stress level increases with the
enhancement of mismatch factors, and it relies on mismatch
factors only from this evidence.
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Fig. 14 Variations of dimensionless open stress under the under
different T-stresses for different creep exponents
7 CONCLUDING REMARKS
A MMBL model is adopted to investigate the influences of
T-stress on constraint effect of creep crack in this paper. The
stress field of creep crack tip in the MMBL model under different
mismatch factors are presented here. The characterization of
constraint effect for creep crack under the mismatch condition is
given. According to the above study, the conclusions can be
obtained as follows:
1) The variations of T-stress in creep regime is presented.
Results show that the T-stress under the same
mismatch factor has the tendency. The stress field of
creep crack in MMBL model is combined by HRR
type field in creep range and elastic field of far field.
The MMBL is valid to be used under the creep regime.
2) The creep zone size of crack is rather different under
different mismatch factors. It implies the creep zone
size on the mismatch condition is affected remarkably
by the mismatch factor. It demonstrates that creep
crack on under-match condition has the larger creep
zone than that on the over-match condition. That
shows the material mismatch with higher constraint
effect can restrict the creep zone, which is very similar
to elastoplastic materials.
3) It can be found that the influence of T-stress on
constraint effect of creep crack within the MMBL
model is quite different from elastoplastic materials.
However, the negative or positive T-stress can still
influence the geometric constraint effect of creep
crack. As usual, the negative T-stress can lead to lower
open stress and has lower constraint effect. Compared
with the material mismatch constraint, the influence of
T-stress on geometric constraint effect seems to be not
significant as expected. The applications of this
conclusion to the specimens or structures need a
further study.
4) Under the over-match condition, there is blunting
effect though the T-stress is under small range. Under
the under-match condition, the influence of T-stress on
open stress is not remarkable if the T-stress is not large
enough. Both positive and negative T-stresses can
accelerate the creep relaxation of creep crack tip field.
Under the mismatch condition, the C(t)-integrals are
really different during short creep time for different
mismatch factors, however, they behave similarly if
the creep time is long enough.
5) The open stresses are the same though the creep
exponents and creep coefficients differ much. It
implies that the level of open stress is dependent on
the mismatch factor for the same loading.
ACKNOWLDGMENT
This work was supported by the National Science Foundation for
Distinguished Young Scholars of China (Grant No. 11325211)
and the Project of International Cooperation and Exchange
NSFC (Grant No. 11511130057).
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[29] Hutchinson, J., 1968, "Singular behaviour at the end of a
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[30] Rice, J., and Rosengren, G., 1968, "Plane strain deformation
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